math.RA — Rings and Algebras

A family of finite dimensional Hopf algebras over $\mathbb{C}$ with varying algebra structure

Contributed by Pavel I Etingof

We construct an algebraic one-parameter family of complex Hopf algebras of dimension $1024$, with fixed coalgebra, whose underlying associative algebras vary in a continuous family of isomorphism classes. Four multiplication lines in the second radical layer of a distinguished block recover the $j$-invariant of the elliptic curve given by the equation $y^2=x(x-1)(x-t^2).$ This answers negatively the finiteness question of Etingof and Gelaki concerning algebra types of Hopf algebras of fixed dimension.